A
couple of months ago I ran across an
article by the mathematician John Allen Paulos
on the ABC News web site, questioning Defense Secretary
Rumsfeld's approach of using small but heavily armed
forces in Iraq instead of the large forces traditionally
required for a military invasion. The article was
written in March 2003, while the conflict in Iraq
was very much under way and the outcome undecided.
Paulos' analysis introduced me to something called
Lanchester's Square Law, a mathematical formula for
calculating the strength of military forces. I've
been interested in this issue for some time, because
I'm curious about the problems of balancing strategy
games, so I decided to find out a little more about
it.

The
Theory of Predictable Victory (or Defeat)

There
are now quite a number of books on the market about
videogame design. To my surprise, I haven't discovered
any that address Lanchester's laws; to my embarrassment,
that includes my own book -- an error that I intend
to rectify if I can persuade my publisher to bring
out a second edition. In the meantime, here's a very
brief introduction to a huge topic called Operations
Research. Theorists of military strategy can stop
reading here; you're not going to learn anything you
don't already know.

Suppose
you have a certain amount of artillery, and your enemy
has a different amount of identical artillery, and
both are steadily shooting at each other and reducing
each other's numbers. If everything else is equal,
who's stronger? Obviously, whoever has more guns.
How much stronger? Obviously, the proportion between
the size of the first army and the size of the second,
right? Blue has three times as many field guns as
red, so blue is three times as strong.

"Wrong!"
said Frederick William Lanchester, a British engineer
and mathematician who lived from the middle 19th century
through the end of World War II. Lanchester was one
of those energetic Victorians who made the British
the masters of the world -- he built the first British
gasoline-powered automobile, invented power steering
and disc brakes, studied problems of aerodynamics,
and more or less created the field of Operations Research
(OR). OR was originally devised to solve problems
associated with logistics, such as moving large quantities
of troops and materiel around, although now it's used
as much for business and industrial simulations as
it is for warfare.

Back
in 1916, during the height of World War I, Lanchester
devised a series of differential equations to demonstrate
the power relationships between opposing forces. Among
these are what is known as Lanchester's Linear Law
(for ancient combat) and Lanchester's Square Law (for
modern combat). In ancient combat, between phalanxes
of men with spears, say, one man could only ever fight
exactly one other man at a time. If each man kills,
and is killed by, exactly one other, then the number
of men remaining at the end of the battle is simply
the difference between the larger army and the smaller,
as you might expect (assuming identical weapons).
This is the "obvious" conclusion I arrived
at above.

In
modern combat, however, with artillery pieces firing
at each other from a distance, the guns can attack
multiple targets and can receive fire from multiple
directions. Lanchester determined that the power of
such an army is proportional not to the number of
units it has, but to the square of the number of units.
This is known as Lanchester's Square Law.

Here's
why. Suppose the blue army is three times the size
of the red army. This means that it is concentrating
three times as much firepower on red as red is firing.
Just as importantly, red's firepower is diluted over
three times as many blue units. The combined effect
of these two conditions is that blue is nine times
as strong as red although it only has three times
as many units. Similarly, the number of units remaining
at the end of the battle is the square root of the
difference between the squares of the numbers of units
on each side. In other words, if blue has five units
and red has three, the number of units left at the
end will be four. (Five squared minus three squared
is 16; its square root is four.)

Squarely
Beaten

This
leads to what Lanchester called his Principle of Concentration:
in modern warfare, bigger armies are better -- in
fact, a lot better than you might think at first.
This may seem trivially obvious, but it has important
strategic consequences. During the Napoleonic Wars,
British naval strategy depended on keeping the French
and Spanish fleets from uniting so that they could
not mass against Britain. Admiral Nelson's victory
at the Battle of Trafalgar depended on splitting up
his enemies' superior force, and destroying the smaller
groups before they could reunite.

Let's
look at another consequence of Lanchester's Square
Law. We in America are very proud of the quality of
our military technology -- what Lanchester would have
called "weapon efficiency," or, more bluntly,
"killing rate." What are the consequences
of one side having better technology than the other?

If
red and blue have equal numbers of units, but red's
weapon efficiency is twice that of blue, then red
is obviously going to win the battle: they can kill
two blue units for every one red unit they lose. But
if blue has three times as many units, the Square
Law still holds in spite of red's technological advantage.
Blue's mathematical advantage is reduced from 9 times
as strong to 4.5 times as strong by red's technology,
but blue still defeats red. In order to compensate
for the other side's numerical advantage, the killing
rate of the technology must advance in proportion
to the square of the number of opponents it is meant
to kill. As Dr. Paulos put it, it takes an N-squared-fold
increase in quality to make up for an N-fold increase
in quantity. That's a tall order.

To
Whom Does the Glory Belong?

Now,
Lanchester's Laws are far from perfect. For one thing,
they only apply to battles of attrition, in which
the object is to wipe out the other side. If "winning"
a battle is defined in some other way -- and modern
Western militaries don't normally consider slaughtering
every last opponent to be a legitimate objective --
then Lanchester's Law has nothing to say about who
wins. Napoleon "won" every battle he fought
in his march into Russia, but he still lost 98 percent
of his men and was forced to retreat without achieving
his objective. Similarly, the British and French dispute
who won a naval engagement called (by the British)
the "Glorious First of June." On June 1,
1794, a British squadron of 26 men-of-war attacked
a French squadron of similar size, sinking or taking
seven ships and driving the rest away in confusion.
The British had a little over 1,000 casualties; the
French, closer to 7,000, so it seems like a British
victory. But the French squadron was protecting a
convoy of 117 cargo ships laden with American grain
bound for starving, revolutionary France. All the
cargo ships escaped and made it to port. So far as
the French are concerned, the heroic sacrifice of
their squadron produced a strategic victory for them.

Lanchester's
Laws also don't take into account such considerations
as terrain, morale, weapon range, movement and maneuver,
surprise, weather, and many other issues that have
decided battles over the centuries. In fact, various
people have argued that in actual battle situations
Lanchester's laws are well nigh useless. (See the
works of Col. Trevor N. Dupuy for further criticism;
Dupuy favored an empirical model of combat based on
analysis of real, historical battles -- not numerical
analysis.) In other words, mathematical models and
computer simulations are all well and good, but because
they don't model actual battle conditions, it's difficult
to know what value they really have to military planners.

A
Fair Fight

However,
to us as game designers, the Lanchester Laws are still
extremely useful. We know we're not modeling actual
battlefield conditions -- certainly not if we're designing
war games about elves and trolls, or robots and aliens.
All we have are mathematical models. All we produce
are computer simulations. It doesn't really matter
if our results don't reflect historical reality, because
historical reality is irrelevant to our purpose.

Questionable
though Lanchester's Laws may be in real-life
military scenarios, it should serve as
a good mathematical model for strategy
game designers in balancing opposing forces
(image from Rise of Nations, Microsoft).

Lanchester's
Laws should be particularly useful in problems like
setting the cost and production rate of different
unit types. Suppose the aliens can produce twice as
many Slime Leeches as the humans can produce Robot
Tanks in a given amount of time, and both sides pay
the same amount for each unit. Suppose further that
the weapons efficiency of a Slime Leech is identical
to that of a Robot Tank. Lanchester's Square Law tells
us that at the end of five minutes of production by
both sides, the aliens will have a force twice as
large, but four times as strong, and costing only
twice as much, as the humans' force. Clearly, the
aliens have a big advantage. To compensate for their
doubled production rate, each Slime Leech should cost
twice as much-in theory, at least. Only playtesting
can determine whether it really works in practice,
given all the other factors in the game.

There's
a great deal of debate about the merits of combat
analysis, and many different theories have emerged
over the years. Logistical, business, and industrial
practices are more suitable for operations research
methods than actual battles, because they're more
rational activities, less influenced by the emotions
of the participants or the charisma of the leaders.
But in war games, we don't usually want to model soldiers'
emotions or leaders' charisma too much anyway -- they
introduce too great a random factor. For our purposes,
operations research, starting with Lanchester's Laws,
seems seems like a valuable tool.